In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infin...In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.展开更多
In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are...In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.展开更多
A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions o...A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.展开更多
The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple ba...The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.展开更多
In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,w...In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].展开更多
We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under som...We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under some suitable assumptions,we prove the existence of a ground state solution of the equation.Additionally,we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.展开更多
In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a ste...In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.展开更多
In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger...In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.展开更多
In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlin...In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.展开更多
This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ+Rψ(x,ψ) in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solution...This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ+Rψ(x,ψ) in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth.展开更多
We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical ...We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.展开更多
We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of...We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of nonnegative ground state solutions is established.Our method relies upon the variational method and some analysis tricks.展开更多
In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence ...In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence of a ground state solution forλ>0 large enough.Furthermore,we verify the asymptotic behavior of ground state solutions asλ→+∞.展开更多
In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent (KGM){-△u+V(x)u-(2w+Φ)Φu=λ|u|^q-2u+|u|^4u,x∈R^3,-△Φ+Φu^2=-wu^2,x∈R^3,where λ and ω are two positive constants....In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent (KGM){-△u+V(x)u-(2w+Φ)Φu=λ|u|^q-2u+|u|^4u,x∈R^3,-△Φ+Φu^2=-wu^2,x∈R^3,where λ and ω are two positive constants. We found the existence of positive ground state solutions (that is, for solutions which minimizes the action functional among all the solutions) of (KGM) which improves some previous existence result in Carriao et al.(2012)[8].展开更多
Silicate perovskites((Mg, Fe)SiO 3 and CaS iO 3) are believed to be the major constituent minerals in the lower mantle. The phase relation, solid solution, spin state of iron and water solubility related to the lo...Silicate perovskites((Mg, Fe)SiO 3 and CaS iO 3) are believed to be the major constituent minerals in the lower mantle. The phase relation, solid solution, spin state of iron and water solubility related to the lower mantle perovskite are of great effect on the geodynamics of the Earth's interior and on ore mineralization. Previous studies indicate that a large amount of iron coupled with aluminum can incorporate into magnesium perovskite, but this is discordant with the disproportionation of(Mg,Fe)SiO 3 perovskite into iron-free MgS i O3 perovskite and hexagonal phase(Mg0.6Fe0.4)SiO 3 in the Earth's lower mantle. MnS iO 3 is the first chemical component confirmed to form wide range solid solution with Ca SiO 3 perovskite and complete solid solution with MgS i O3 perovskite at the P-T conditions in the lower mantle, and addition of Mn Si O3 will strongly affects the mutual solubility between Mg Si O3 and CaS iO 3. The spin state of iron is deeply depends on the site occupation of the Fe3+or Fe2+, the synthesis and the annealing conditions of the sample. It seems that the spin state of Fe2+ in the lower mantle perovskite can be settled as high spin, however, the existence of intermediate spin or low spin state of Fe2+ in perovskite has not been clarified. Moreover, different results have also been reported for the spin state of Fe3+ in perovskite. The water solubility of the lower mantle perovskite is related with its composition. In pure Mg SiO 3 perovskite, only less than 500 ppm water was reported. Al–Mg Si O3 perovskite or Al–Fe–MgS iO 3 perovskite in the lower mantle accommodates water of 1100 to 1800 ppm. Further experiments are necessary to clarify the detailed conditions for perovskite solid solution, to reliably analyze the valence and spin states of iron in the coexisting iron-bearing phases, and to compare the water solubility of different phases at different layers for deeply understanding the geodynamics of the Earth's interior and ore mineralization.展开更多
The convergent iterative procedure for solving the groundstate Schrodinger equation is extended to derive the excitation energy and the wavefunction of the low-lying excited states. The method is applied to the one-di...The convergent iterative procedure for solving the groundstate Schrodinger equation is extended to derive the excitation energy and the wavefunction of the low-lying excited states. The method is applied to the one-dimensional quartic potential problem. The results show that the iterative solution converges rapidly when the coupling g is not too small.展开更多
The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potenti...The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials.As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.展开更多
Using results from various reactions that populate 10He, I conclude that the ground state has E2n = 1.07(7) MeV and the excited 0+ state is in the region of 2.1-3.1 MeV. The amount of the (sd)2 component in the g...Using results from various reactions that populate 10He, I conclude that the ground state has E2n = 1.07(7) MeV and the excited 0+ state is in the region of 2.1-3.1 MeV. The amount of the (sd)2 component in the ground state is less than about 0.075.展开更多
Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform a...Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.展开更多
文摘In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.
基金supported by the NSFC (12071438)supported by the NSFC (12201232)
文摘In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.12261064 and 11861048)the Natural Science Foundation of Inner Mongolia,China (Grant Nos.2021MS01004 and 2022QN01008)the High-level Talents Scientific Research Start-up Foundation of Inner Mongolia University (Grant No.10000-21311201/165)。
文摘A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.
文摘The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.
基金Supported by the National Natural Science Foundation of China (11971393).
文摘In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].
基金supported by National Natural Science Foundation of China(11971202)Outstanding Young foundation of Jiangsu Province(BK20200042)。
文摘We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under some suitable assumptions,we prove the existence of a ground state solution of the equation.Additionally,we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.
基金the National Natural Science Foundation of China (11971393)。
文摘In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.
文摘In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.
基金supported by the National NaturalScience Foundation of China(12071170,11961043,11931012,12271196)supported by the excellent doctoral dissertation cultivation grant(2022YBZZ034)from Central China Normal University。
文摘In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.
基金supported by the Hunan Provincial Innovation Foundation for Postgraduate(CX2013A003)the NNSF(11171351,11361078)SRFDP(20120162110021)of China
文摘This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ+Rψ(x,ψ) in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth.
基金the Science and Technology Project of Education Department in Jiangxi Province(GJJ180357)the second author was supported by NSFC(11701178).
文摘We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.
基金National Natural Science Foundation of China(11471267)the first author was supported by Graduate Student Scientific Research Innovation Projects of Chongqing(CYS17084).
文摘We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of nonnegative ground state solutions is established.Our method relies upon the variational method and some analysis tricks.
基金supported by National Natural Science Foundation of China(11971393)。
文摘In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence of a ground state solution forλ>0 large enough.Furthermore,we verify the asymptotic behavior of ground state solutions asλ→+∞.
基金partially supported by the National Natural Science Foundation of China(11801400,11571187)partially supported by the National Natural Science Foundation of China(11861053)the Postdoctoral Science Foundation of China(2017M611159)
文摘In this article, we study the following Klein-Gordon-Maxwell system involving critical exponent (KGM){-△u+V(x)u-(2w+Φ)Φu=λ|u|^q-2u+|u|^4u,x∈R^3,-△Φ+Φu^2=-wu^2,x∈R^3,where λ and ω are two positive constants. We found the existence of positive ground state solutions (that is, for solutions which minimizes the action functional among all the solutions) of (KGM) which improves some previous existence result in Carriao et al.(2012)[8].
基金partly supported by projects from JSPS KAKENHI (Grant No. 18340167)MEXT KAKENHI (Grant No. 20103002)+2 种基金NSFC (Grand No.90914002)China Geological Survey (Grant No. 1212011220926)the Ministry of Education of China (Grant No. 20130022110003)
文摘Silicate perovskites((Mg, Fe)SiO 3 and CaS iO 3) are believed to be the major constituent minerals in the lower mantle. The phase relation, solid solution, spin state of iron and water solubility related to the lower mantle perovskite are of great effect on the geodynamics of the Earth's interior and on ore mineralization. Previous studies indicate that a large amount of iron coupled with aluminum can incorporate into magnesium perovskite, but this is discordant with the disproportionation of(Mg,Fe)SiO 3 perovskite into iron-free MgS i O3 perovskite and hexagonal phase(Mg0.6Fe0.4)SiO 3 in the Earth's lower mantle. MnS iO 3 is the first chemical component confirmed to form wide range solid solution with Ca SiO 3 perovskite and complete solid solution with MgS i O3 perovskite at the P-T conditions in the lower mantle, and addition of Mn Si O3 will strongly affects the mutual solubility between Mg Si O3 and CaS iO 3. The spin state of iron is deeply depends on the site occupation of the Fe3+or Fe2+, the synthesis and the annealing conditions of the sample. It seems that the spin state of Fe2+ in the lower mantle perovskite can be settled as high spin, however, the existence of intermediate spin or low spin state of Fe2+ in perovskite has not been clarified. Moreover, different results have also been reported for the spin state of Fe3+ in perovskite. The water solubility of the lower mantle perovskite is related with its composition. In pure Mg SiO 3 perovskite, only less than 500 ppm water was reported. Al–Mg Si O3 perovskite or Al–Fe–MgS iO 3 perovskite in the lower mantle accommodates water of 1100 to 1800 ppm. Further experiments are necessary to clarify the detailed conditions for perovskite solid solution, to reliably analyze the valence and spin states of iron in the coexisting iron-bearing phases, and to compare the water solubility of different phases at different layers for deeply understanding the geodynamics of the Earth's interior and ore mineralization.
基金This research was supported in part by the U.S. Department of Energy (Grant No DE-FG02-92ER-40699) and the National Natural Science Foundation of China (Grant No 10547001).
文摘The convergent iterative procedure for solving the groundstate Schrodinger equation is extended to derive the excitation energy and the wavefunction of the low-lying excited states. The method is applied to the one-dimensional quartic potential problem. The results show that the iterative solution converges rapidly when the coupling g is not too small.
基金*Supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2010291, the Professor and Doctor Foundation of Yancheng Teachers University under Grant No. 07YSYJB0203
文摘The newly developed iterative method based on Green function defined by quadratures along a single trajectory is combined with the variational method to solve the ground state quantum wave function for central potentials.As an example, the method is applied to discuss the ground state solution of Yukawa potential, using Hulthen solution as the trial function.
文摘Using results from various reactions that populate 10He, I conclude that the ground state has E2n = 1.07(7) MeV and the excited 0+ state is in the region of 2.1-3.1 MeV. The amount of the (sd)2 component in the ground state is less than about 0.075.
文摘Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.